( ) G. Narsimlu Department of Mathematics, Chaitanya Bharathi Institute of Technology, Gandipet, Hyderabad, India

Transcription

1 VOL. 3, NO. 9, OCTOBER 8 ISSN ARPN Jornal of Engneerng and Appled Scences 6-8 Asan Research Pblshng Networ (ARPN). All rghts reserved. SOLUTION OF AN UNSTEADY FLOW THROUGH POROUS MEDIA PAST ON MOVING VERTICAL PLATE WITH VARIABLE TEMPERATURE AND HEAT SOURCE IN THE PRESENCE OF INCLINED MAGNETIC FIELD IN THE PRESENCE OF VISCOUS DISSIPATION G. Narsml Department of Mathematcs, Chatana Bharath Insttte of Technolog, Gandpet, Hderabad, Inda E-Mal: gnr.cbt@gmal.com ABSTRACT In ths paper we analze and std the problem of an nstead flow throgh poros meda past on movng vertcal plate wth varable temperatre and heat sorce n the presence of nclned magnetc feld and vscos dsspaton s stded. The governng eqatons are dscrtzed b Galern fnte-element method and are solved b Cran-Ncolson method sng C-Program.The effects of varable parameters to the veloct and temperatre profles are dscssed graphcall and the nmercal vales obtaned for sn-frcton have been tablated. Kewords: nclned magnetc feld, vscos dsspaton, heat sorce, poros meda, fnte-element method. INTRODUCTION The problem of free convecton flows past a vertcal srface has been stded extensvel becase of ts wde applcaton n man branches of engneerng and scences. Power generaton sstems, coolng of nclear reactors, Geophscs, agrcltre and thermal nslaton. Rapt and Kmar [] have analzed MHD flow past an mplsvel started movng vertcal plate wth varable temperatre and mass dffson. Seth and Ghosh [] stded an nstead hdro-magnetc flow n a rotatng channel n the presence of nclned magnetc feld. The effect of temperatre dependent vscost and thermal condctvt on nstead MHD convectve heat transfer past a sem-nfnte vertcal poros plate has been stded b Sedde and Salama [3]. Chamha AJ et al [4] presented nstead MHD free convecton flow past an exponentall accelerated vertcal plate wth mass transfer, chemcal reacton and thermal radaton. R.N. Bar [5] stded free convecton heat and mass transfer MHD Flow n a vertcal channel n the presence of the chemcal reacton. Hossan M. A. [6] analzed the effect on of hall crrent on nstead hdromagnetc freeconvecton flow near an nfnte vertcal poros plate. Mohmed Abd E-Azz [7] presented nstead mxed convecton heat transfer along a vertcal stretchng srface wth varable vscost and vscos dsspaton. U.S Rapt and S. Kmar [8] have stded radaton effects on MHD flow past an mplsvel started vertcal plate wth varable heat and mass transfer. A. Ogl [9] stded the nflence of radaton absorpton on nstead free convecton and mass transfer flow n the presence of a nform magnetc feld. T. Arnmar and L Anand Bab [] have analzed the std of Radaton effect of MHD flow past an mplsve started vertcal plate wth varable temperatre and nform mass dffson throgh A fnte element method The obectve of ths paper s to std abot the nstead flow throgh poros meda past on movng vertcal plate wth varable temperatre and heat sorce n the presence of nclned magnetc feld and vscos dsspaton. FORMULATION OF THE PROBLEM Consder an nstead MHD flow of a vscos ncompressble radatng fld b a vertcal non condctng plate and the plate mplsvel started movng wth veloct tang nto accont n the presence of vscos dsspaton and heat sorce. A nform magnetc feld B s assmed to be appled on the plate wth angle. Intall the plate and the fld are at same temperatret. At tme t, temperatre of the plate s ncreased tot. The governng eqatons of flow w feld are as nder follows: Momentm eqaton: æ ( ) sn ö sb a v = n + gb T - T t + - r ç çè ø Energ eqaton: æ ö T T m Q = + t rcp rc p + - rc ç è ø p ( T T ) where s the veloct components n, g s the gravtatonal acceleraton, b s the thermal expanson coeffcent, T s the thermal temperatre nsde the thermal bondar laer, s the thermal condctvt, r s the fld denst, s the coeffcent of vscost, K s the permeablt of the medm C s the specfc heat at constant pressre, s s the electrc condctvt, T s p () () 873

2 VOL. 3, NO. 9, OCTOBER 8 ISSN ARPN Jornal of Engneerng and Appled Scences 6-8 Asan Research Pblshng Networ (ARPN). All rghts reserved. the temperatre of the fld far awa from the plate, s tme. The bondar condtons are: t - t : =, T = T " üï ï ïïï t (3) t > : =, T = T + ( Tw - T ) at = ý v ïïïï T T as ïþ Where s the veloct of the fld, o Tw and Cw are the temperatre and concentraton of the wall respectvel, s the tme. Introdcng the followng non- dmensonal qanttes, t n vc r t vq müï =, =, =,Pr =, K =, t =, Q =, v = ï v v v C r ïïý p n r p ( w - ) T - T gbv T T s B l v ï q =, Gr =, Ha =, l =, Ec =, 3 Tw - T m cp( Tw - T ) ï ïþ (4) where Gr, Pr, Ha, Ec, l and Q are the thermal Grashof nmber, Prandtl nmber, Hartmann nmber, Ecert nmber, characterstc length scale and heat absorpton parameter respectvel. Wth the help of non- dmensonal qanttes, eqatons () and () becomes t K Gr M Ec Q t Pr Here M Ha sn condton are (5) (6) The correspondng dmensons less bondar t : =, q = " üï ïï t > : =, q = t at = ý ïï q as ïþ METHOD OF SOLUTION On solvng the eqatons (5) and (6). B applng Galern fnte element method for eqaton (5) over the element ) s: (e, (7) ( et ) e N N P d (8) t Where P ( Gr), N M ntegratng the frst term n K eqaton (8) b parts and neglectng the frst term weget T N ( et ) e N N Pd t N Let = be the lnear pecewse approxmaton solton over the element (e) ) where ( N N N N T, and, N are the bass fnctons. On smplfng we get ( ) ( ) ( ) e e e l Nl Pl l 6 6 where dot are denotes dfferentaton wth respect to. Assemblng the element eqatons for two consectve elements and followng s obtaned: N P 4 4 l 6 6 (9) Now pt row correspondng to the node to zero, from eqaton () the dfference schemes wth ( l e ) h s: 4 N 4 P 6 h 6 Applng Cran - Ncholson method to the above eqaton, we get 874

3 VOL. 3, NO. 9, OCTOBER 8 ISSN ARPN Jornal of Engneerng and Appled Scences 6-8 Asan Research Pblshng Networ (ARPN). All rghts reserved. A + A + A = A4 - + A5 + A6 + + P () where A A 6r N A A 6r N , A r N A r N P P Gr Now from eqaton (6) followng eqatons obtaned: B B B B B B P Where 5 () B B3 Pr6r QPr, B4 B6 Pr 6r QPr, P P Pr Ec B 4Prr 4Q Pr, B 4Prr 4Q Pr, Here, r and h, are mesh sze along the h drecton and the tme drecton respectvel. Index refers to the space, and refers to the tme. In the Eqatons () - (), tang =,..., n and sng bondar condtons (7), the followng sstem of eqatons are obtaned: Pr =.7, α = 3, K =., Ha = 4, t =., Q =, Ec =. Gr =, 5, Fgre-. Effect of Gr on veloct profles. Pr =.7, Gr =, α = 3, Ha = 4, t =., Q =, Ec =. K =.,.,.4 A X = B, =,..., n () Where A s are matrx of order n and X, B s colmn matrces havng n components. The soltons of above sstems of eqatons are obtaned b sng the Thomas algorthm for veloct, temperatre and concentraton. Also the nmercal soltons are obtaned b exectng the C-program wth the smaller vales of h and. No sgnfcant change was observed n, and θ then the Galern fnte element method s stable and convergent. The dmenson less sn frcton obtaned as RESULTS AND DISCUSSIONS The nmercal vales were compted wth respect to the phscal parameters, lethermal Grashofnmber Gr, permeablt K, Ecert nmber Ec, Prandtl nmber Pr, tme t, angle of nclnaton of magnetc feld α, Hartmann nmber Ha, and heat sorce parameter as shown n fgres Fgre-. Effect of K on veloct profles. Fgres and llstrate the behavor of the veloct for dfferent vales of the Gr and K. The nmercal reslts show that the effect of ncreasng vales of Grashof nmber and permeablt reslts n a decreasng veloct. From fgre 3 t s observed that an ncreasng n the Ecert nmber Ec leads to rse n the vales of veloct. Fgres 4 and 5 represent the veloct and temperatre profles for dfferent vales of the Prandtlnmber Pr. It s notced that the veloct and temperatre decrease as there ncrease n the Pr vales Fgre-3. Effect of Ec on veloct profles. 875

4 VOL. 3, NO. 9, OCTOBER 8 ISSN ARPN Jornal of Engneerng and Appled Scences 6-8 Asan Research Pblshng Networ (ARPN). All rghts reserved. t =.,.5,..3.. Pr =.7, Gr =, α = 3, K =., Ha = 4, t =., Q =, Ec = Fgre-4. Effect of Pr on veloct profles.. Fgre-7. Effect of t on temperatre profles. Pr =.7, Gr =, α = 3, K =., Ha = 4, Ec =.,t = Q =.,.5, Fgre-8. Effect of Q on veloct profles. Fgre-5. Effect of Pr on temperatre profles. Fgre-9. Effect of on veloct profle. Fgre-6. Effect of t on veloct profles. 876

5 VOL. 3, NO. 9, OCTOBER 8 ISSN ARPN Jornal of Engneerng and Appled Scences 6-8 Asan Research Pblshng Networ (ARPN). All rghts reserved. Fgres 9 and llstrate the behavor of the veloct and temperatre for dfferent vales of angle of nclnaton of magnetc feld α and Hartmann nmber (Ha). It s observed that an ncrease n α and Haleads to a decrease n the vales of veloct and temperatre Pr =.7, Gr =, α = 3, K =., Ha = 4, t=., Ec =. Q =.,.5, Fgre-. Effect of Ha on veloct profles. Fgres 6 and 7 represent the veloct and temperatre profles for dfferent vales of t. As the tme ncreases, the veloct and temperatre ncrease. The effect of the heat sorce parameter on the veloct and temperatre are shown n Fgres 8 and. As heat sorce parameter ncreases, veloct and temperatre ncreases Fgre-. Effect of Q on temperatre profles. Table vales of Sn frcton. Pr Gr Ha K α Q Ec t CONCLUSIONS In ths paper a nmercal std has been done to std an nstead flow throgh poros meda past on movng vertcal plate wth varable temperatre and heat sorce n the presence of nclned magnetc feld and vscos dsspaton. Reslts for the model have been derved b Galarn fnte element technqe. Some conclsons of the std are as below: Veloct ncreases wth the ncrease n thermal Grashofnmber Gr, permeablt K, Ecert nmber Ec, heat sorce parameter Q and tme t. Veloct decreases wth the ncrease n Prandtlnmber Pr, an angle of nclnaton of magnetc feld α, Hartmann nmber Ha. Temperatre decreases wth the ncrease n Prandtlnmber Pr. Temperatre ncreases wth the ncrease n tme t and heat sorce parameter Q. Sn frcton decreases wth the ncrease Hartmann nmber Ha. 877

6 VOL. 3, NO. 9, OCTOBER 8 ISSN ARPN Jornal of Engneerng and Appled Scences 6-8 Asan Research Pblshng Networ (ARPN). All rghts reserved. REFENCES [] U. S. Rapt and Srendra Kmar.. MHD Flow past an Implsvel Started Vertcal Plate wth Varable Temperatre and Mass Dffson. Appled Mathematcal Scences. 5(3): nform mass Dffson - A fnte element method. Ind. J. Sc. and tech. (3): 3-9. [] Seth GS. and Ghosh SK Unstead hdromagnetc flow n a rotatng channel n the presence of nclned magnetc feld. Int. J. of Engneerng Sc. 4(7): [3] Seddee M. A. and Salama F. A. 7. The effect of temperatre dependent vscost and thermal condctvt on nstead MHD convectve heat transfer past a sem-nfnte vertcal poros plate. J. comptatonal Materals Sc. 4: [4] Chamha A.J., M.C Ra., T. Sdhaar Redd and S.V.K Varma. 4. Unstead MHD free convecton flow past an exponentall accelerated vertcal plate wth mass transfer, chemcal reacton and thermal radaton. Int. J. of Mcroscale and Nanoscale thermal. 5(): [5] R.N.Bar. 3. Free convecton heat and mass transfer MHD Flow n a vertcal channel n the presence of the chemcal reacton. Internatonal Jornal of analss and applcaton. 3(): 5-8. [6] Hossan M. A Effect of hall crrent on nstead hdromagnetc free-convecton flow near an nfnte vertcal poros plate. Jornal of the Phscal Socet of Japan. 55(7): [7] Mohmed Abd E-Azz. 4. Unstead mxed convecton heat transfer along a vertcal stretchng srface wth varable vscost and vscos dsspaton. Jornal of the Egptan Mathematcal Socet. : [8] U.S Rapt and S. Kmar.. Radaton effects on MHD flow past an mplsvel started vertcal plate wth varable heat and mass transfer. Int. J. of Appl. Math. And Mech. 8(): [9] A. Ogl. 5. The nflence of radaton absorpton on nstead free convecton and mass transfer flow n the presence of a nform magnetc feld. Internatonal Jornal of heat and mass transfer. 48(3): [] Arnmar and L Anand Bab. 3. Std of Radaton effect of MHD flow past an mplsve started vertcal plate wth varable temperatre and 878